Reduction and Bifurcation of Traveling Waves of the Kdv-Burgers-Kuramoto Equation

DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2016036 DYNAMICAL SYSTEMS SERIES B Volume 21, Number 6, August 2016 pp. 2057–2071

REDUCTION AND BIFURCATION OF TRAVELING WAVES OF THE KDV-BURGERS-KURAMOTO EQUATION

Yuqian Zhou School of Applied Mathematics, Chengdu University of Information Technology Chengdu, Sichuan 610225, China Qian Liu∗ School of Computer Science and Technology, Southwest University for Nationalities Chengdu, Sichuan 610041, China

(Communicated by Shigui Ruan)

Abstract. In this paper, the Lie symmetry analysis is performed on the KBK equation. By constructing its one-dimensional optimal system, we obtain four classes of reduced equations and corresponding group-invariant solutions. Par- ticularly, the traveling wave equation, as an important reduced equation, is 3 investigated in detail. Treating it as a singular perturbation system in R , we study the phase space geometry of its reduced system on a two-dimensional invariant manifold by using the dynamical system methods such as tracking the unstable manifold of the saddle, studying the equilibria at inﬁnity and dis- cussing the homoclinic bifurcation and Poincar´ebifurcation. Correspongding wavespeed conditions are determined to guarantee the existence of various bounded traveling waves of the KBK equation.

1. Introduction. This paper considers the KdV-Burgers-Kuramoto (KBK) equation [14] ∂u ∂u ∂2u ∂3u ∂4u + u +α ¯ + β¯ +γ ¯ = 0, (1) ∂t ∂x ∂x2 ∂x3 ∂x4 which is an appropriate model to describe phenomena that are simultaneously involved in nonlinearity, dissipation, dispersion and instability, whereα, ¯ β,¯ γ¯ are nonzero real constants. Equation (1) plays important roles in describing physical processes in motion of turbulence and unstable systems [30, 17, 18] and is also known as the generalized Kuramoto-Sivashinsky equation [17] or Benney equation [25]. The KBK equation has been investigated widely and various direct methods have been proposed to obtain exact traveling wave solutions of it, such as the Weiss-Tabor-Carnevale transformation method [17], trial-function method [25], tanh-function method and extended tanh-function method [19, 28,4]. In recent decade, more methods are applied to obtain new exact solutions of it, including the trigonometric function expansion method [6], generalized F-expansion method [34], uniﬁed ans¨atzeapproach [15], a combination method [32] and Exp-function

2010 Mathematics Subject Classiﬁcation. Primary: 58F15, 58F17; Secondary: 53C35. Key words and phrases. Lie symmetry analysis, traveling waves, invariant manifold, geometric singular perturbation, KdV-Burgers-Kuramoto equation. ∗ Corresponding author: Qian Liu.

2057 2058 YUQIAN ZHOU AND QIAN LIU method [16, 24]. In addition, some numerical methods for solving the fractional KBK equation can also be found in [29, 31, 10]. Though there have been so many profound results about traveling wave solutions of the KBK equation which contributed to our understanding of nonlinear physical phenomena and wave propagation, some traveling wave solutions could be still lost because of defects of the direct methods caused by auxiliary equations and hypotheses about the forms of solutions. Not only that, it is well known that not even all traveling wave solutions can be expressed analytically. So, qualitative analysis of traveling waves of the KBK equation is still necessary and significant. In addition, we also want to explore whether there exist other types of solutions of the KBK equation except the traveling wave solutions, for example, the series solutions. These problems arouse our great interest in surveying the KBK equation again. In order to obtain new solutions of the KBK equation, we need to reduce it as simple as possible. It is known that the Lie group method is a powerful approach to reduce nonlinear partial differential equations and construct their group-invariant solutions. The fundamental basis of the technique is that, when a differential equation is invariant under a Lie group of transformations, a reduction transformation exists. Most of the required theory and description of the techniques of this method can be found in [26, 27] as well as in [11,1,2]. By using the Lie symmetry analysis method [26], we obtain finite-dimensional symmetries of the KBK equation, which form a three-dimensional Lie algebra. Subsequently, a one-dimensional optimal system for the KBK equation is given according to the adjoint representation of vector field. Based on the optimal system, we get four classes of reduced equations of the KBK equation and corresponding group-invariant solutions which are not reported before. Particularly, as an important reduced equation, the traveling wave equation of the KBK equation is investigated qualitatively in detail. Our strategy is to transform it to a three-dimensional differential dynamical system with singular perturbation. This idea is firstly introduced by Fu and Liu [7] in 2010. From geometric singular perturbation point of view, they give the existence condition of kink waves by proving that a strictly increasing traveling front of the KdV-Burgers equation can persist in the KBK equation for sufficiently small dissipation parameterγ ¯. But, interactions and combination of nonlinearity, dissipation, dispersion and instability could lead to more complicated behaviors of traveling waves, such as the periodic motion, oscillation. It means that other types of bounded traveling waves could occur to the KBK equation besides the traveling fronts. In fact, three basic types of bounded traveling waves could occur for a PDE, which are periodic waves, kink waves (or shock waves) and solitary waves. Recall that heteroclinic orbits are trajectories which have two distinct equilibria as their α and ω-limit sets and homoclinic orbits are trajectories whose α and ω-limit sets consist of the same equilibrium. So, the three basic types of bounded traveling waves mentioned above correspond to periodic, heteroclinic and homoclinic orbits of the traveling wave system of a PDE respectively [9, 13, 20, 21, 22]. It is just the relation that offers an effective way to study traveling waves of a PDE from the point of view of differential dynamical system. So, the dynamical system methods paly important roles in our paper, which is the most remarkable difference from [7]. It will be seen that this method allows detailed analysis on phase space geometry of traveling wave system of the KBK REDUCTION AND BIFURCATION OF TRAVELING WAVES 2059 equation so that all possible bounded traveling waves and corresponding existence conditions can be identified clearly. 3 We first prove that it has a two-dimensional invariant submanifold M in R by the geometric singular perturbation theory [5, 12]. Restricted on M, the singular perturbation system is reduced to a dynamical system with perturbation in R2. By using the dynamical system methods and some techniques such as tracking unstable manifold of the saddle and studying equilibria at infinity, we investigate the phase space geometry of the corresponding unperturbed system in detail. The result clearly shows that under appropriate wavespeed conditions there exist the saddle- spiral shock waves besides the traveling fronts mentioned in [7] for the unperturbed system. Subsequently, by using the Fredholm theorem, we prove that the two types of traveling waves can persist in the KBK equation. Furthermore, Mel’nikov type computation is carried out to study the homoclinic bifurcation and Poincaré bifurcation of the reduced system. Various wavespeed conditions are determined to guarantee the existence of solitary waves and periodic waves of the KBK equation. Not only that, based on the bifurcation results, we also discuss the existence of two types of oscillatory bounded traveling waves for the KBK equation.

2. Reduction and series solutions of the KBK equation. To seek a symmetry σ(x, t, u) of the KBK equation, we set

σ = a(x, t)ut + b(x, t)ux + d(x, t)u + e(x, t), where u = u(x, t) satisfies equation (1) and a(x, t), b(x, t), d(x, t), e(x, t) are functions to be determined later. Based on Lie group theory [26], σ satisfies the following equation ¯ σt + uxσ + uσx +ασ ¯ xx + βσxxx +γσ ¯ xxxx = 0. (2) ¯ Noting that ut = −uux − αu¯ xx − βuxxx − γu¯ xxxx, we can calculate the expressions of σt, σx, σxx, σxxx and σxxxx (see Appendix 1 ). Substituting them into (2), we get the differential equations with respect to a(x, t), b(x, t), d(x, t) and e(x, t). Solving them yields

a(x, t) = c1, b(x, t) = c2t + c3, d(x, t) = 0, e(x, t) = −c2, where c1, c2, c3 are arbitrary real constants. Thus, the symmetry of the KBK equation can be written as

σ = c1ut + c2(tux − 1) + c3ux. It follows that the vector ﬁeld of the KBK equation is spanned by the vectors ∂ ∂ ∂ ∂ V = ,V = ,V = t + . 1 ∂t 2 ∂x 3 ∂x ∂u Next, we construct the one-dimensional optimal system for the KBK equation. Firstly, one can check that the following commutation relations hold

[V1,V2] = 0, [V1,V3] = V2, [V2,V3] = 0. (3) It means that the vector ﬁeld is closed under the Lie bracket. From the commutation relations (3), we calculate adjoin representations of the vector ﬁeld as follows

Ad(exp(εVi))Vi = Vi, i = 1, 2, 3, 2060 YUQIAN ZHOU AND QIAN LIU

Ad(exp(εV1))V2 = V2, Ad(exp(εV1))V3 = V3 − εV2,

Ad(exp(εV2))V1 = V1, Ad(exp(εV2))V3 = V3,

Ad(exp(εV3))V1 = V1 + εV2, Ad(exp(εV3))V2 = V2 for any ε ∈ R. Given a nonzero vector

V = a1V1 + a2V2 + a3V3, we expect to simplify as many of the coeﬃcients ai as possible through suitable applications of adjoint maps to V . Firstly, suppose that a3 6= 0. Without loss of generality, we can set a3 = 1. If we act on the vector V by Ad(exp(a2V1)), the coeﬃcient of the V2 can be eliminated:

V¯ = Ad(exp(a2V1))V = a1V1 + V3. It is easy to see that V¯ can not be reduced further by above adjoint maps. So, every one-dimensional subalgebra generated by V with a3 6= 0 is equivalent to the subalgebra spanned by a1V1 + V3(a1 6= 0) or V3(a1 = 0). The remaining one-dimensional subalgebras are spanned by vectors V with a3 = 0 i.e., V = a1V1 + a2V2. Similarly, suppose that a1 = 1. If we act on the vector V by Ad(exp(−a2V3)), the coeﬃcient of the V2 can be eliminated:

V¯ = Ad(exp(−a2V3))V = V1.

So, every one-dimensional subalgebra generated by V with a3 = 0 is equivalent to the subalgebra spanned by V1(a1 6= 0) or V2(a1 = 0). Thus , we obtain an optimal system of one-dimensional subalgebras of the KBK equation as follows:

{V1,V2,V3, a1V1 + V3} where a1 6= 0 is an arbitrary constant. According to the optimal system, we will reduce the KBK equation and construct its group-invariant solutions.

Case 1. For the generator V2, we have

u(x, t) = f1(t). (4) Substituting (4) into (1), we reduce the KBK equation to the form: 0 f1 = 0,

0 df1 where f1 = dt . Solving it, we get the trivial solution of the KBK equation u(x, t) = C, where C is an arbitrary constant.

Case 2. For the generator V3, we have −1 u(x, t) = f2(t) + xt . (5) Substituting (5) into (1), we reduce the KBK equation to the form: 0 tf2 + f2 = 0,

0 df2 where f2 = dt . Solving it, we get the trivial solution of the KBK equation u(x, t) = Ct−1 + xt−1, where C is an arbitrary constant.

Case 3. For the generator V1, we have

u(x, t) = f3(x). (6) REDUCTION AND BIFURCATION OF TRAVELING WAVES 2061

Substituting (6) into (1), we reduce the KBK equation to the form: 0000 ¯ 000 00 0 γf¯ 3 + βf3 +αf ¯ 3 + f3f3 = 0, (7)

0 df3 where f3 = dx . Integrating (7) once, we have 1 γf¯ 000 + βf¯ 00 +αf ¯ 0 + f 2 + e = 0, (8) 3 3 3 2 3 where e is an integral constant. We assume a solution of Eq.(8) in a power series of the from

∞ X n f3(x) = cnx . (9) n=0 Substituting (9) into (8) leads to

∞ P n ¯ 6¯γc3 +γ ¯ (n + 3)(n + 2)(n + 1)cn+3x + 2βc2 n=1

∞ ∞ ¯ P n P n +β (n + 2)(n + 1)cn+2x +αc ¯ 1 +α ¯ (n + 1)cn+1x (10) n=1 n=1

∞ n 1 2 1 P P n + 2 c0 + 2 ( ckcn−k)x + e ≡ 0. n=1 k=0 From (10), we have −1 1 c = (2βc¯ +αc ¯ + c2 + e) (11) 3 6¯γ 2 1 2 0 and n ¯ 1 P −(β(n + 2)(n + 1)cn+2 +α ¯(n + 1)cn+1 + 2 ckcn−k) c = k=0 , (12) n+3 γ¯(n + 3)(n + 2)(n + 1) for n ≥ 1 Thus, the power series solution of the KBK equation can be written as follows

∞ 2 3 P n+3 u(x, t) = c0 + c1x + c2x + c3x + cn+3x n=1

2 1 ¯ 1 2 3 = c0 + c1x + c2x − 6¯γ (2βc2 +αc ¯ 1 + 2 c0 + e)x (13)

n ¯ 1 P ∞ β(n+2)(n+1)cn+2+¯α(n+1)cn+1+ 2 ckcn−k P k=0 n+3 − γ¯(n+3)(n+2)(n+1) x , n=1 where ci(i = 0, 1, 2) are arbitrary constants, the other coeﬃcients cn(n ≥ 3) are determined by (11) and (12). By a simple application of the Implicit Function Theorem, it is easy to check that power series (13) is convergent.

Case 4. For the generator a1V1 + V3, we have t u(x, t) = f4(ξ) + , (14) a1 2062 YUQIAN ZHOU AND QIAN LIU

2 where ξ = x − t . Substituting (14) into (1), we reduce the KBK equation to the 2a1 form: 0000 ¯ 000 00 0 1 γf¯ 4 + βf4 +αf ¯ 4 + f4f4 + = 0, (15) a1

0 df4 where f4 = dξ . Similarly, we can obtain the corresponding series solution of Eq. (15) as follows 2 2 2 t t t 2 t 3 u(x, t) = + c0 + c1(x − ) + c2(x − ) + c3(x − ) a1 2a1 2a1 2a1 (16) ∞ t2 4 P t2 n+3 +c4(x − ) − cn+3(x − ) , 2a1 2a1 n=2 −1 ¯ 1 2 where ci(i = 0, 1, 2) are arbitrary constants, c3 = 6¯γ (2βc2 +αc ¯ 1 + 2 c0 + e), c4 = 1 ¯ 1 − (6βc3 + 2¯αc2 + c1c0 + ) and other coeﬃcients cn are determined by cn+3 = 24¯γ a1 n ¯ 1 P −(β(n+2)(n+1)cn+2+¯α(n+1)cn+1+ 2 ckcn−k) k=0 γ¯(n+3)(n+2)(n+1) , (n ≥ 2). Especially, the traveling wave solutions correspond to the symmetry group generated by Ad(exp(−cV¯ 3))V1 = V1 +cV ¯ 2.

For the generator V1 +cV ¯ 2, we have u(x, t) = f(ξ) = f(x − ct¯ ), which convert the KBK equation into its traveling wave system γf¯ 0000 + βf¯ 000 +αf ¯ 00 + ff 0 − cf¯ 0 = 0, (17) where 0 denotes d/dξ and constantc ¯ > 0 is wavespeed.

3 3. Existence of invariant submanifold M in R and the ﬂow on it. Inte- grating (17) once will yield 1 γf¯ 000 + βf¯ 00 +αf ¯ 0 + f 2 − cf¯ = 0, (18) 2 where we set the integration constant e = 0 (Otherwise, we can always make a suitable homeomorphic transformation to eliminate it and convert the equation into the same form as equation (18) ). Firstly, we rescale the parameterγ ¯ = γ for the small > 0. Then, the homeomorphic transformation f(ξ) = γU(ξ) convert (18) into the form 1 U 000 + βU 00 + αU 0 + U 2 − cU = 0, 2 β¯ α¯ c¯ where β = γ , α = γ , c = γ . The equivalent system of it is  0  U = V, V 0 = Y, (19)  0 1 2 Y = cU − 2 U − αV − βY. With ξ = τ, the ‘fast system’ associated with (19) has the form  0  U = V, V 0 = Y, (20)  0 1 2 Y = cU − 2 U − αV − βY, REDUCTION AND BIFURCATION OF TRAVELING WAVES 2063

0 where denotes d/dτ.(20) has two equilibria E0 : (0, 0, 0) and E1 : (2c, 0, 0) at which the Jacobian matrices are  0 0   0 0   0 0   0 0  J(E0) =   and J(E1) =   c −α −β −c −α −β with corresponding characteristic equations ∆(x) − c2 = 0 and ∆(x) + c2 = 0 3 2 1 respectively, where ∆(x) = x +β x +α x. Function ∆(x) has three zeros 0, − 2 β ± 1 p 2 1 d∆(x) 2 β − 4 a in C . Noting that dx |x=0 6= 0 , we can see that J(E0) and J(E1) have at least a pair nonzero real eigenvalues with opposite signs. We want to prove that system (19) has two manifolds intersecting along an one-dimensional curve in R3. This curve just corresponds to a bounded traveling wave solution of the KBK equation. If is set to zero in (19), then U and V are governed by U 0 = V, (21) V 0 = Y, where Y lies on the set 1 M = (U, V, Y ) ∈ 3 : cU − U 2 − αV − βY = 0 0 R 2 which is a two-dimensional submanifold in R3. From [5], the manifold M0 is said to be normally hyperbolic if the linearization of the fast system, restricted to M0, has exactly dimM0 eigenvalues on the imaginary axis, with the remainder of the system hyperbolic. The ‘fast system’ (20), restricted to the manifold M0, has the Jacobian matrix  0 0 0   0 0 0    c − U −a −β which has the eigenvalues 0, 0, −β. It means that the manifold M0 is normally hyperbolic. So, Fenichels invariant manifold theory [5] guarantees that there exists 3 a two-dimensional submanifold M diffeomorphic to M0 in R , which is within the distance ε of M0 and is invariant for the flow (19). Next, we assume that the manifold M can be written as 3 M = (U, V, Y ) ∈ R : Y = h(U, V, ) , cU αV U 2 where function h(U, V, ) satisfies h(U, V, 0) = β − β − 2β . In order to obtain the approximation of manifold M, we expand function h(U, V, ) in Taylor series in the variable cU αV U 2 h(U, V, ) = − − + h (U, V, 0) + O(2). (22) β β 2β 1 Substituting (22) into (19), we get α cU α2 + β c V α U 2 UV h1(U, V, 0) = − − + β3 β3 2β3 β2 2064 YUQIAN ZHOU AND QIAN LIU by power of . This allows one to write (19) as the following system ( U 0 = V, 0 cU αV U 2 2 (23) V = β − β − 2β + h1(U, V, 0) + O( ), which determines the dynamics on the ‘slow’ manifold M. Letting = 0 in (23), we have ( U 0 = V = P (U, V ), 0 cU αV U 2 (24) V = β − β − 2β = Q(U, V ),

0 0 which has two equilibria E0 : (0, 0) and E1 : (2c, 0), at which the Jacobian matrices are " 0 1 # " 0 1 # 0 0 J(E0) = c α and J(E1) = c α β − β − β − β √ √ 2 2 0 −α± α +4 β c 0 −α± α −4 β c with the eigenvalues λ1,2(E0) = 2β and λ1,2(E1) = 2β respectively. Without loss of generality, we only consider the case α > 0, β > 0 for (24). In fact, all other cases can be converted to the case by suitable homeomorphic transformations. For instance, if α < 0, β > 0, we can make the homeomorphic transformation V = −Ve , ξ = −τ. If α > 0, β < 0, we can make the homeomorphic transformation U = −Ue + 2c, ξ = −τ. Similarly, if α < 0, β < 0, we can make the homeomorphic transformation U = −Ue + 2c, V = −Ve. For the case α > 0, β > 0, 0 0 2 E0 is a saddle and E1 is a stable node when α ≥ 4βc or a stable focus when 2 0 α < 4βc. We will prove that there exists a heteroclinic orbit connecting E0 and 0 E1. Theorem 3.1. Suppose that α > 0, β > 0. Then system (24) has a saddle-node α2 α2 heteroclinic orbit when c ≤ 4β or a saddle-focus heteroclinic orbit when c > 4β . Proof. Note that the expression α ∂P (U, V )/∂U + ∂Q(U, V )/∂V = − β has a fixed sign. By the Bendixon Theorem, system (24) has no closed orbit in phase plane (U-V plane). α2 When α > 0, β > 0 and c ≤ 4β , we consider the problem in the triangle region D 2 D := {(U, V ) ∈ R : 0 < U < 2c, 0 < V < k(U − 2c)} which is enclosed by three lines U = 0,V = 0 and V = k(U − 2c) in phase plane of (24), where k < 0 is a constant to be determined. In region D, there is no equilibrium of (24). Firstly, by [3], there exists an unstable manifold Γ of the saddle 0 E0 in the first quadrant, which intersects neither the U-axis nor the V -axis in an 0 enough small neighborhood of the origin E0. The vector field defined by (24) guarantees that orbits confined to the first quadrant move to the right as ξ increases. It means that Γ can not intersect the 2 boundary ∂D1 := {(U, V ) ∈ R : U = 0, 0 ≤ V ≤ −2kc}. On the boundary 2 dV cU U 2 ∂D2 := {(U, V ) ∈ R : 0 < U < 2c, V = 0}, dξ |∂D2 = β − 2β > 0, which means REDUCTION AND BIFURCATION OF TRAVELING WAVES 2065 that Γ can not intersect the boundary ∂D2. On the boundary ∂D3 := {(U, V ) ∈ R2 : 0 < U < 2c, V = k(U − 2c)}, we have dV F (U) α | = − , dU ∂D3 βk(U − 2c) β 1 2 where F (U) = cU − 2 U . From the fact F (2c) = 0, F (U) F (U) − F (2c) = > F 0(2c) = −c, U − 2c U − 2c dV F (U) α −c α which leads to dU |∂D3 = βk(U−2c) − β < βk − β . So, there exists negative constant k satisfying −c − α ≤ k since α2 ≥ 4βc. To be more specific, we can choose the βk β √ √ −α− α2−4βc −α+ α2−4βc constant k in the interval ( 2β , 2β ) freely to guarantee Γ not to intersect the boundary ∂D3. From the facts above, it concludes that Γ can not go out of the region D and 0 E1 is exactly the ω-limit set of it. Thus, we prove the existence of saddle-node 0 0 heteroclinic orbit connecting E0 and E1. Moreover, the fact dU/dξ = V > 0 means that the bounded kink wave solution corresponding to Γ is monotone increasing with respect to ξ. α2 When α > 0, β > 0 and c > 4β , we need to consider the problem globally. With the Poincarétransformation U = 1/y, V = x/y and dτ = dξ/y ,(24) can be changed into 0 1 cy α xy 2 x = − 2β + β − β − x y, y0 = −xy2, which has no equilibrium in (x, y)-plane when y = 0. Then by another Poincarétransformation U = x/y, V = 1/y and dτ = dξ/y, (24) can be changed into 0 x = y + P2(x, y), 0 y = Q2(x, y), which has an equilibrium (0, 0) corresponding to the equilibrium at infinity in V - α xy cx2y x3 α y2 cxy2 x2y axis, where P2(x, y) = β − β + 2β ,Q2(x, y) = β − β + 2β . (0, 0) is a degenerate equilibrium with nilpotent Jacobian matrix. So, we need more precise analysis for it. By implicit function theorem, we can solve the equation y + p2(x, y) = 0 in an enough small neighborhood of the origin (0, 0) and obtain x3 α x4 y = φ(x) = − + + O(x5). 2β 2β2 Let 5 6 x α x 7 Ψ(x) := Q2(x, φ(x)) = − + + O(x ), 4β2 2β3 ∂P (x, φ(x)) ∂Q (x, φ(x)) 2x2 3α x3 δ(x) := 2 + 2 = − + O(x4). ∂x ∂y β 2β2 By Theorem 7.2 and its corollary in [35], the degenerate equilibrium (0, 0) is an unstable degenerate node. So, we can give the global phase portrait of system (24) in figure1. It shows that there exists a saddle-focus heteroclinic orbit connecting 0 0 E0 and E1, which corresponds to the saddle-spiral shock wave. 2066 YUQIAN ZHOU AND QIAN LIU

Figure 1. Global phase portrait of (24) for α > 0, β > 0 and α2 < 4βc.

Assume that the heteroclinic connection of (24) can be expressed as (U0,V0). The one thing left is to prove there exists analogous heteroclinic connection for (23) when is suﬃciently small. In order to seek such connection in (23), we set

U = U0 + U,V˜ = V0 + V.˜ (25) Substituting (25) into (23) and taking the lowest order in , we obtain the approx- imate system U˜ 0 L = (26) V˜ h1(U0,V0, 0) which governs U˜ and V˜ , where the linear diﬀerential operator is deﬁned by " # U˜ d U˜ 0 1 U˜ L = − . V˜ dξ V˜ c−U0 α V˜ β − β Next, we want to prove system (26) has a solution satisfying U,˜ V˜ → 0 as ξ → ±∞. By Fredholm theory [33], system (26) has a square-integrable solution if and only if the following compatibility condition holds Z +∞ 0 X(ξ), dξ = 0 (27) −∞ h1(U0(ξ),V0(ξ), 0) for all functions X(ξ) in the kernel of the adjoint operator L∗, where h·, ·i is the inner product on R2. The adjoint system for (26) can be expressed as

" c−U0 # dX 0 − = β X. (28) dξ α −1 β Noting that ξ → +∞, U0 √→ 2c, we can see that (28) has a constant matrix with α± α2−4 β c two eigenvalues λ1,2 = 2β . Obviously, both eigenvalues λ1 and λ2 have positive real parts. Any solutions of (28), other than the zero solution, must grow REDUCTION AND BIFURCATION OF TRAVELING WAVES 2067 exponentially. The only solution in L2 is therefore a zero solution X(ξ) = 0, and consequently the Fredholm orthogonality condition (27) trivially holds. Thus, we prove the existence of analogous heteroclinic orbits for (23) when is suﬃciently small, which implies the desired existence of kink wave and saddle-spiral shock wave for the KBK equation.

4. Solitary wave and periodic wave solutions. In this section, under the condition that α > 0 and β > 0, we will prove the existence of homoclinic and periodic orbits of (23), which corresponds to the existence of solitary wave and periodic wave solutions of the KBK equation respectively. Rescale the parameter α = α˜. System (23) becomes ( U 0 = V, 0 cU U 2 (29) V = β − 2β + G(U, V, ), (˜αβ+c)V VU ˜ where G(U, V, ) = − β2 + β2 +O(). When = 0, (29) has a saddle E0 : (0, 0) and a center E˜1 : (2c, 0). In fact, in this case, (29) is a Hamiltonian system with the ﬁrst integral 1 cU 2 U 3 H(U, V ) := V 2 − + . (30) 2 2β 6β

By the properties of Hamiltonian system, there exists a homoclinic orbit Υ0 deter- 1 2 cU 2 U 3 mined by the level curve 2 V − 2β + 6β = 0. In the compact region enclosed by homoclinic loop Υ ∪ {E˜ }, there is a family of periodic orbits Γ (h) surrounding 0 0 E˜1 the center E˜1 (see ﬁgure2), where 3 2 2c Γ ˜ (h) := {(U, V ) ∈ R : H(U, V ) = h, h ∈ (− , 0)}. E1 3β

Figure 2. Global phase portrait of (29) for c = 1, β = 1 and = 0. Theorem 4.1. Suppose that α = α˜ > 0, β > 0 and 0 < 1 . 7 1. For arbitrary , there exists a c = c()(c(0) = 5 αβ˜ ) to guarantee that system (29) has a homoclinic orbit. 7 2. System (29) has a periodic orbit when αβ˜ < c < 5 αβ˜ . 2068 YUQIAN ZHOU AND QIAN LIU

Proof. Consider the Mel’nikov function M(h, α,˜ β, c) = H G(U, V, 0)dU = H (− (˜αβ+c)V + VU )dU Γ Γ β2 β2 (31) I0 = β2 (−αβ˜ − c + R(h)), H k where Ik := Γ U V dU, k = 0, 1, R(h) := I1/I0 and Γ corresponds to homoclinic loop Υ ∪ {E˜ } or periodic orbits Γ (h). 0 0 E˜1 First, we claim that lim R(h) = 2c, (32) 2c3 h→− 3β 12 lim R(h) = c. (33) h→0 7 In fact, assume that (a(h), 0) and (b(h), 0) (a(h) < b(h)) are the points where periodic orbit Γ (h) intersects the U-axis . Noting that b(h) − a(h) → 0 when E˜1 2c3 h → − 3β , we have R b(h) I (h) UV dU lim R(h) = lim 1 = lim a(h) = 2c, 3 3 3 b(h) h→− 2c h→− 2c I0(h) h→− 2c R 3β 3β 3β a(h) V dU implying (32). Note the facts that homoclinic orbit Υ0 intersects U-axis at the point (3c, 0) and periodic orbits Γ (h) approach the homoclinic orbit Υ as h → 0. Thus E˜1 0 ∗ ∗ lim R(h) = lim I1(h)/I0(h) = I1 /I0 , where h→0 h→0 ∗ H R 3c I := I0(0) = V dU = 2 V (U) dU, 0 Υ0∪{E˜1} 0 ∗ H R 3c I := I1(0) = UV dU = 2 UV (U) dU, 1 Υ0∪{E˜1} 0 √ −3 β (−3 c+U)U and V (U) = 3β , U ∈ (0, 3c) is the explicit expression of the curve Υ0. ∗ ∗ 12 Direct computation gives the result I1 /I0 = 7 c and thus proves (33). From the fact that I0 > 0 is the area of the region enclosed by Γ, the Mel’nikov function has 7 a simple zero c = 5 αβ˜ . By [8], we know that for every sufficiently small , there 7 exists a corresponding c = c()(c(0) = 5 αβ˜ ) to guarantee that the homoclinic orbit survive from homoclinic bifurcation and therefore prove the existence of homoclinic orbit. In order to prove the result of periodic orbit in theorem 4.1, we need to investigate Poincarébifurcation for system (29). Firstly, we claim that function R(h) 2c3 is monotone for h ∈ (− 3β , 0). Our strategy is to use Theorem 2 in [23] to prove it. Note that the energy function (30) has the form of variable separation. By Theorem 2 in [23], in order to prove the monotonicity of R(h), it suffices to verify the following conditions (LZ1): G˜0(U)(U − 2c) > 0(or < 0) for U ∈ (0, 3c)\{2c}, (LZ2): f1(U)f1(U˜) > 0 for U ∈ (0, 2c), (LZ3): ζ0(U) < 0(or > 0) for U ∈ (0, 2c), ˜ cU 2 U 3 ˜ ˜ ˜ ˜ ˜ where G(U) := − 2β + 6β , U = U(U) is defined by G(U) = G(U), fi(U) = U i−1, i = 1, 2, and f (U)G˜0(U˜) − f (U˜)G˜0(U) ζ(U) := 2 2 . (34) 0 0 f1(U)G˜ (U˜) − f1(U˜)G˜ (U) REDUCTION AND BIFURCATION OF TRAVELING WAVES 2069

2 ˜0 U(U−2 c) Condition (LZ1) can be verified since G (U)(U − 2c) = 2β > 0 for U ∈ (0, 3c)\{2c}. Condition (LZ2) holds naturally. Furthermore, from (34) one can calculate M˜ (U,U˜ ) + M˜ (U, U˜) dU˜ ζ0(U) = dU , (35) (U˜ − 2c + U)2 where M˜ (U, U˜) := U(U − 2c). Note that M˜ (U,U˜ ) > 0, M˜ (U, U˜) < 0 and dU˜ G0(U) U(U − 2c) = = < 0 dU G0(U˜) U˜(U˜ − 2c) for 0 < U < 2c < U˜ < 3c. It follows from (35) that ζ0(U) > 0 for U ∈ (0, 2c). 0 2c3 This concludes that R (h) < 0 for h ∈ (− 3β , 0) by Theorem 2 in [23]. Thus, the monotonicity of R(h) is proved. 7 ∗ The monotonicity of R(h) implies that ifαβ ˜ < c < 5 αβ˜ , there exists a h ∈ 2c3 ∗ ∗ (− 3β , 0) satisfies M(h , α,˜ β, c) = 0. Furthermore, h is a simple zero of the 0 ∗ 0 ∗ I0(h ) ∗ Mel’nikov function M(h, α,˜ β, c), since M (h , α,˜ β, c) = β2 (−αβ˜ − c + R(h )) + ∗ ∗ I0(h ) 0 ∗ I0(h ) 0 ∗ β2 R (h ) = β2 R (h ) < 0. Therefore, by the Poincarébifurcation theory [3], 7 whenαβ ˜ < c < 5 αβ˜ and is small enough, there exists a limit cycle for system (29), which is close to the periodic orbit Γ (h∗). E˜1 5. Existence of other bounded traveling waves. In section 3 and 4, we obtain different wavespeed conditions to guarantee the existence of three basic types of bounded traveling waves of the KBK equation. From these bifurcation results, existence of more bounded traveling waves of the KBK equation can be identified. In fact, one can note that the region enclosed by a periodic orbit or a homoclinic loop is compact. It means that orbits in the compact region are bounded, and therefore correspond to the bounded traveling waves of the KBK equation. In addition, from properties of the dynamical system, it is well known that the limit cycle and the homoclinic loop are limit sets. So, when a limit cycle appears from the Poincare bifurcation, there will exist some connections between the equilibrium and the limit cycle, which correspond to a kind of oscillatory bounded traveling wave. Similarly, if a homoclinic orbit persists from the homoclinic bifurcation, there will exist some connections between the equilibrium and the homoclinic loop, which corresponds to another kind of oscillatory bounded traveling wave.

Acknowledgments. This work is supported by the Natural Science Foundation of China (No.11301043 and No.11171046), Innovative Research Team of the Educa- tion Department of Sichuan Province (15TD0050), the Key Project of Educational Commission of Sichuan Province (No.12ZA224), the Scientiﬁc Research Foundation of CUIT (No.J201219).

Appendix 1. ¯ ¯ 2 σt = −atα¯ ux,x − atuux − atβ ux,x,x − atγ¯ ux,x,x,x + 3 aβ ux,x 2 ¯2 2 −buux,x + au ux,x − bα¯ ux,x,x + aβ ux,x,x,x,x,x + 2 aux u 2 2 −bγ¯ ux,x,x,x,x + aα¯ ux,x,x,x − dγ¯ ux,x,x,x + aγ¯ ux,x,x,x,x,x,x,x ¯ ¯ −bβ ux,x,x,x − dα¯ ux,x − dβ ux,x,x + btux + dtu + et ¯ +5 auxβ ux,x,x + 6 auxγ¯ ux,x,x,x + 2 auα¯ ux,x,x ¯ ¯ +2 auβ ux,x,x,x + 2 aα¯ β ux,x,x,x,x + 2 aα¯ γ¯ ux,x,x,x,x,x 2070 YUQIAN ZHOU AND QIAN LIU

¯ +2 aβ γ¯ ux,x,x,x,x,x,x + 10 aγ¯ ux,xux,x,x + 4 auxα¯ ux,x 2 −bux − duux + 2 auγ¯ ux,x,x,x,x

¯ 2 σx = −axuux − axα¯ ux,x − axβ ux,x,x − axγ¯ ux,x,x,x − aux ¯ −auux,x − aα¯ ux,x,x − aβ ux,x,x,x − aγ¯ ux,x,x,x,x + bxux +bux,x + dxu + dux + ex ¯ σxx = −ax,xuux − ax,xα¯ ux,x − ax,xβ ux,x,x − ax,xγ¯ ux,x,x,x 2 ¯ −2 axux − 2 axuux,x − 2 axα¯ ux,x,x − 2 axβ ux,x,x,x −2 axγ¯ ux,x,x,x,x − 3 auxux,x − auux,x,x − aα¯ ux,x,x,x ¯ −aβ ux,x,x,x,x − aγ¯ ux,x,x,x,x,x + bx,xux + 2 bxux,x +bux,x,x + dx,xu + 2 dxux + dux,x + ex,x ¯ σxxx = −3 ax,xα¯ ux,x,x − ax,x,xβ ux,x,x − ax,x,xα¯ ux,x 2 −ax,x,xγ¯ ux,x,x,x − ax,x,xuux − 3 aux,x + bx,x,xux +3 bx,xux,x + 3 bxux,x,x + bux,x,x,x + dx,x,xu + 3 dx,xux 2 +3 dxux,x + dux,x,x − 3 ax,xux − 9 axuxux,x − 3 axuux,x,x ¯ −3 axβ ux,x,x,x,x − 3 axα¯ ux,x,x,x − 3 axγ¯ ux,x,x,x,x,x ¯ −4 auxux,x,x − auux,x,x,x − aα¯ ux,x,x,x,x − aβ ux,x,x,x,x,x ¯ −aγ¯ ux,x,x,x,x,x,x − 3 ax,xβ ux,x,x,x − 3 ax,xuux,x −3 ax,xγ¯ ux,x,x,x,x + ex,x,x ¯ σxxxx = −ax,x,x,xβ ux,x,x − ax,x,x,xα¯ ux,x − 6 ax,xα¯ ux,x,x,x ¯ −4 ax,x,xβ ux,x,x,x − 4 ax,x,xuux,x − 4 ax,x,xα¯ ux,x,x −4 ax,x,xγ¯ ux,x,x,x,x − 18 ax,xuxux,x − 6 ax,xuux,x,x ¯ −16 axuxux,x,x − 6 ax,xβ ux,x,x,x,x − 6 ax,xγ¯ ux,x,x,x,x,x ¯ −4 axα¯ ux,x,x,x,x − 4 axuux,x,x,x − 4 axβ ux,x,x,x,x,x −4 axγ¯ ux,x,x,x,x,x,x − 5 auxux,x,x,x − aγ¯ ux,x,x,x,x,x,x,x ¯ −auux,x,x,x,x − aα¯ ux,x,x,x,x,x − aβ ux,x,x,x,x,x,x −ax,x,x,xγ¯ ux,x,x,x − 10 aux,xux,x,x − ax,x,x,xuux +ex,x,x,x + bx,x,x,xux + 4 bx,x,xux,x + 6 bx,xux,x,x +4 bxux,x,x,x + bux,x,x,x,x + dx,x,x,xu + 4 dx,x,xux 2 +6 dx,xux,x + 4 dxux,x,x + dux,x,x,x − 12 axux,x 2 −4 ax,x,xux

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